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nimDividedCohomology -- Computes the dimension or character of sheaf cohomology of twists of divided powers of the cotangent sheaf on projective space, in characteristic 2

Description

This function computes in characteristic 2 the sheaf cohomology of twists of divided powers of the cotangent sheaf on projective space $H^i(\mathbb{P}^{n-1}, D^d \mathcal{R}(d))$, where $D^d \mathcal{R}$ is the d-th divided power of the universal rank (n-1) subsheaf $\mathcal{R}$.

i1 : i =1; d = 5; e = 4; n = 3;
i5 : nimDividedCohomology(i,d,e,n)

o5 = 8

This function can also output the character instead of just the dimension, setting the option FindCharacter to be true.

i6 : i = 1; d = 5; e = 4; n = 3;
i10 : nimDividedCohomology(i,d,e,n, FindCharacter => true)

       4 3 2    3 4 2    4 2 3     3 3 3    2 4 3    3 2 4    2 3 4
o10 = x x x  + x x x  + x x x  + 2x x x  + x x x  + x x x  + x x x
       1 2 3    1 2 3    1 2 3     1 2 3    1 2 3    1 2 3    1 2 3

o10 : ZZ[x ..x ]
          1   3

This function allows the user to control the ambient ring of the character, using the ambient ring R as the fourth input instead then n.

i11 : i = 1; d = 5; e = 4; R = ZZ[x_1..x_3];
i15 : nimDividedCohomology(i,d,e,R)

       4 3 2    3 4 2    4 2 3     3 3 3    2 4 3    3 2 4    2 3 4
o15 = x x x  + x x x  + x x x  + 2x x x  + x x x  + x x x  + x x x
       1 2 3    1 2 3    1 2 3     1 2 3    1 2 3    1 2 3    1 2 3

o15 : R

Ways to use nimDividedCohomology:

  • nimDividedCohomology(ZZ,ZZ,ZZ,PolynomialRing)
  • nimDividedCohomology(ZZ,ZZ,ZZ,ZZ)

For the programmer

The object nimDividedCohomology is a method function with options.


The source of this document is in /build/reproducible-path/macaulay2-1.25.06+ds/M2/Macaulay2/packages/IncidenceCorrespondenceCohomology.m2:1692:0.